K-theory of the $C^*$-algebra of multivariable Wiener-Hopf operators associated with some polyhedral cones in $R^n$

Abstract

We consider th $C^*$-algebra $W H(R^n,P)$ of the multivariable Wiener-Hopf operators associated with a polyhedral cone in $R^n$ and the extension $0 \rightarrow \mathcal{K} \rightarrow W H(R^n,P) \rightarrow W H (R^n,P)/\mathcal{K} \rightarrow 0$. The main theorem states that if $P$ satisfies suitable geometric conditions (satisfied, e.g., for all simplicial cones and the cones in $R^n,n \leq 3$), then $K_*(W H(R^n,P)) = (0,0); K_*(W H(R^n,P)/\mathcal{K}) = (0,Z)$ and that the index map is an isomorphism. In the cource of the proof we construct a Fredholm operator in $W H(R^n, P)$ with an index 1. The proof is inductive and uses the Mayer-Vietoris exact sequence and the standart six term exact sequence in $K$-theory.